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5 Trigonometric Identities 5.1 Fundamental Identities 5.2 Verifying Trigonometric Identities 5.3 Sum and Difference Identities for Cosine 5.4 Sum and Difference Identities for Sine and Tangent 5.5 Double-Angle Identities 5.6 Half-Angle Identities Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 5-1 5.1 Trigonometric Identities Fundamental Identities ▪ Using the Fundamental Identities Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 5-2 5.1 Example 1 Finding Trigonometric Function Values Given One Value and the Quadrant If value. and is in quadrant IV, find each function (a) In quadrant IV, Copyright © 2008 Pearson Addison-Wesley. All rights reserved. is negative. 5-3 5.1 Example 1 Finding Trigonometric Function Values Given One Value and the Quadrant (cont.) If value. and is in quadrant IV, find each function (b) (c) Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 5-4 5.1 Example 2 Expressing One Function in Terms of Another Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 5-5 5.1 Example 3 Rewriting an Expression in Terms of Sine and Cosine Write in terms of then simplify the expression. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. and , and 5-6 5.2 Verifying Trigonometric Identities Verifying Identities by Working With One Side ▪ Verifying Identities by Working With Both Sides Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 5-7 5.2 Example 1 Verifying an Identity (Working With One Side) Verify that is an identity. Left side of given equation Right side of given equation Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 5-8 5.2 Example 2 Verifying an Identity (Working With One Side) Verify that is an identity. Simplify. 5-9 5.2 Example 3 Verifying an Identity (Working With One Side) Verify that is an identity. Simplify. Factor. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 5-10 5.2 Example 4 Verifying an Identity (Working With One Side) Verify that is an identity. Multiply by 1 in the form Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 5-11 5.2 Example 5 Verifying an Identity (Working With Both Sides) Verify that identity. is an Working with the left side: Multiply by 1 in the form Simplify. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 5-12 5.2 Example 5 Verifying an Identity (Working With Both Sides) (cont.) Working with the right side: Factor the numerator. Distributive property. Factor the denominator. Simplify. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 5-13 5.2 Example 5 Verifying an Identity (Working With Both Sides) (cont.) So, the identity is verified. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 5-14 5.3 Sum and Difference Identities for Cosine Difference Identity for Cosine ▪ Sum Identity for Cosine ▪ Cofunction Identities ▪ Applying the Sum and Difference Identities Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 5-15 5.3 Example 1 Finding Exact Cosine Function Values Find the exact value of each expression. (a) Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 5-16 5.3 Example 1 Finding Exact Cosine Function Values (cont.) (b) (c) Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 5-17 5.3 Example 2 Using Cofunction Identities to Find θ Find an angle θ that satisfies each of the following. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 5-18 5.3 Example 3 Reducing cos (A – B) to a Function of a Single Variable Write cos(90° + θ) as a trigonometric function of θ alone. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 5-19 5.3 Example 2B Using Cofunction Identities to Find θ (Miscellaneous HW Examples) Find an angle θ that satisfies each of the following. 1. Sin (θ + 15o) = Cos (2θ + 5o) Now see 5.3 # 38 2. Write Cos π/12 as cofunction Now see 5.3 #18 5-20 5.3 Example 4 Finding cos (s + t) Given Information About s and t Suppose that , and both s and t are in quadrant IV. Find cos(s – t). The Pythagorean theorem gives Since s is in quadrant IV, y = –8. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 5-21 5.3 Example 4 Finding cos (s + t) Given Information About s and t (cont.) Use a Pythagorean identity to find the value of cos t. Since t is in quadrant IV, Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 5-22 5.3 Example 4 Finding cos (s + t) Given Information About s and t (cont.) Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 5-23 5.4 Sum and Difference Identities for Sine and Tangent Sum and Difference Identities for Sine ▪ Sum and Difference Identities for Tangent ▪ Applying the Sum and Difference Identities Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 5-24 5.4 Example 1 Finding Exact Sine and Tangent Function Values Find the exact value of each expression. (a) Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 5-25 5.4 Example 1 Finding Exact Sine and Tangent Function Values (cont.) (b) Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 5-26 5.4 Example 1 Finding Exact Sine and Tangent Function Values (cont.) (c) Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 5-27 5.4 Example 2 Writing Functions as Expressions Involving Functions of θ Write each function as an expression involving functions of θ. (a) (b) (c) Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 5-28 5.4 Example 4 Verifying an Identity Using Sum and Difference Identities Verify that the equation is an identity. Combine the fractions. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 5-29 5.4 Example 4 Verifying an Identity Using Sum and Difference Identities (cont.) Expand the terms. Combine terms. Factor. So, the identity is verified. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 5-30 5.5 Double-Angle Identities Double-Angle Identities ▪ Omit Product-to-Sum and Sum-toProduct Identities Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 5-31 5.5 Example 1 Finding Function Values of 2θ Given Information About θ The identity for sin 2θ requires cos θ. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 5-32 5.5 Example 1 Finding Function Values of 2θ Given Information About θ (cont.) Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 5-33 5.5 Example 1 Finding Function Values of 2θ Given Information About θ (cont.) Alternatively, find tan θ and then use the tangent double-angle identity. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 5-34 5.5 Example 2 Finding Function Values of θ Given Information About 2θ Find the values of the six trigonometric functions of θ if Use the identity to find sin θ: θ is in quadrant III, so sin θ is negative. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 5-35 5.5 Example 2 Finding Function Values of θ Given Information About 2θ (cont.) Use the identity to find cos θ: θ is in quadrant III, so cos θ is negative. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 5-36 5.5 Example 2 Finding Function Values of θ Given Information About 2θ (cont.) Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 5-37 5.6 Half-Angle Identities Half-Angle Identities ▪ Applying the Half-Angle Identities Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 5-38 5.6 Example 1 Using a Half-Angle Identity to Find an Exact Value Find the exact value of sin 22.5° using the half-angle identity for sine. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 5-39 5.6 Example 2 Using a Half-Angle Identity to Find an Exact Value Find the exact value of tan 75° using the identity Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 5-40 5.6 Example 3 Finding Function Values of s Given 2 Information About s The angle associated with is positive while Copyright © 2008 Pearson Addison-Wesley. All rights reserved. lies in quadrant II since are negative. 5-41 5.6 Example 3 Finding Function Values of s Given 2 Information About s (cont.) Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 5-42 6 Inverse Circular Functions and Trigonometric Equations 6.1 Inverse Circular Functions 6.2 Trigonometric Equations I 6.3 Trigonometric Equations II 6.4 Equations Involving Inverse Trigonometric Functions Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 6-43 6.1 Inverse Circular Functions Inverse Functions ▪ Inverse Sine Function ▪ Inverse Cosine Function ▪ Inverse Tangent Function ▪ Remaining Inverse Circular Functions ▪ Inverse Function Values Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 6-44 6.1 Example 1 Finding Inverse Sine Values Find y in each equation. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 6-45 6.1 Example 1 Finding Inverse Sine Values (cont.) Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 6-46 6.1 Example 1 Finding Inverse Sine Values (cont.) is not in the domain of the inverse sine function, [–1, 1], so does not exist. A graphing calculator will give an error message for this input. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 6-47 6.1 Example 2 Finding Inverse Cosine Values Find y in each equation. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 6-48 6.1 Example 2 Finding Inverse Cosine Values Find y in each equation. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 6-49 6.1 Example 3 Finding Inverse Function Values (DegreeMeasured Angles) Find the degree measure of θ in each of the following. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 6-50 6.1 Example 4 Finding Inverse Function Values With a Calculator (a) Find y in radians if With the calculator in radian mode, enter as y = 1.823476582 Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 6-51 6.1 Example 4(b) Finding Inverse Function Values With a Calculator (b) Find θ in degrees if θ = arccot(–.2528). A calculator gives the inverse cotangent value of a negative number as a quadrant IV angle. The restriction on the range of arccotangent implies that the angle must be in quadrant II, so, with the calculator in degree mode, enter arccot(–.2528) as Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 6-52 6.1 Example 4(b) Finding Inverse Function Values With a Calculator (cont.) θ = 104.1871349° Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 6-53 6.1 Example 5 Finding Function Values Using Definitions of the Trigonometric Functions Evaluate each expression without a calculator. Since arcsin is defined only in quadrants I and IV, and is positive, θ is in quadrant I. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 6-54 6.1 Example 5(a) Finding Function Values Using Definitions of the Trigonometric Functions (cont.) Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 6-55 6.1 Example 5(b) Finding Function Values Using Definitions of the Trigonometric Functions Since arccot is defined only in quadrants I and II, and is negative, θ is in quadrant II. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 6-56 6.1 Example 5(b) Finding Function Values Using Definitions of the Trigonometric Functions (cont.) Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 6-57 6.1 Example 6(a) Finding Function Values Using Identities Evaluate the expression without a calculator. Use the cosine difference identity: Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 6-58 6.1 Example 6(a) Finding Function Values Using Identities (cont.) Sketch both A and B in quadrant I. Use the Pythagorean theorem to find the missing side. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 6-59 6.1 Example 6(a) Finding Function Values Using Identities (cont.) Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 6-60 6.1 Example 6(b) Finding Function Values Using Identities Evaluate the expression without a calculator. sin(2 arccot (–5)) Let A = arccot (–5), so cot A = –5. Use the double-angle sine identity: Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 6-61 6.1 Example 6(b) Finding Function Values Using Identities (cont.) Sketch A in quadrant II. Use the Pythagorean theorem to find the missing side. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 6-62 6.1 Example 6(b) Finding Function Values Using Identities (cont.) Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 6-63 6.2 Trigonometric Equations I Solving by Linear Methods ▪ Solving by Factoring ▪ Solving by Quadratic Methods ▪ Solving by Using Trigonometric Identities Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 6-64 6.2 Example 1 Solving a Trigonometric Equation by Linear Methods is positive in quadrants I and III. The reference angle is 30° because Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 6-65 6.2 Example 1 Solving a Trigonometric Equation by Linear Methods (cont.) Solution set: {30°, 210°} Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 6-66 6.2 Example 2 Solving a Trigonometric Equation by Factoring or Solution set: {90°, 135°, 270°, 315°} Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 6-67 6.2 Example 3 Solving a Trigonometric Equation by Factoring Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 6-68 6.2 Example 3 Solving a Trigonometric Equation by Factoring (cont.) has two solutions, the angles in quadrants III and IV with the reference angle .729728: 3.8713 and 5.5535. has one solution, Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 6-69 6.3 Trigonometric Equations II Equations with Half-Angles ▪ Equations with Multiple Angles Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 6-70 6.3 Example 1 Solving an Equation Using a Half-Angle Identity (a) over the interval and (b) give all solutions. is not in the requested domain. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 6-71 6.3 Example 2 Solving an Equation With a Double Angle Factor. or Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 6-72 6.3 Example 3 Solving an Equation Using a Multiple Angle Identity From the given interval 0° ≤ θ < 360°, the interval for 2θ is 0° ≤ 2θ < 720°. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 6-73 6.3 Example 3 Solving an Equation Using a Multiple Angle Identity (cont.) Since cosine is negative in quadrants II and III, solutions over this interval are Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 6-74 6.4 Equations Involving Inverse Trigonometric Functions Solving for x in Terms of y Using Inverse Functions ▪ Solving Inverse Trigonometric Equations Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 6-75 6.4 Example 1 Solving an Equation for a Variable Using Inverse Notation Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 6-76 6.4 Example 2 Solving an Equation Involving an Inverse Trigonometric Function Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 6-77 6.4 Example 3 Solving an Equation Involving Inverse Trigonometric Functions Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 6-78 6.4 Example 3 Solving an Equation Involving Inverse Trigonometric Functions (cont.) Sketch u in quadrant I. Use the Pythagorean theorem to find the missing side. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 6-79